Fibonacci Numbers, Calculating The Golden Ratio with Sequences , Measuring with Proportional Calipers

Lauren SiegelMaking Math, Teacher Support & Training1 Comment

One derivation of the golden ratio or phi uses the geometry of the pentagon.

Mark your elbow, wrist and tip of longest finger – The calipers allow you to compare this ratio to phi and also to that of a honey bee’s body.

We used the “stone” path to introduce the sequences that associate with Fibonacci numbers and with geometric progressions. We can derive phi by comparing the two sequences. We can also play a Fibonacci game on the path.


The Measurement Activities (Mr. Morales/ Mr. Ahrens).  In the planetarium We’ll have the giant calipers, the arm – wrist – hand proportional measurement tool, some pictures of various animals, the Hurricane Sandy Puzzle with its golden spiral, the Golden Ratio Hand Mirrors, and the calipers to give to each student.  Students will also visit the specimen room in the next building to explore, use their calipers and chat with Mr. Ahrens about some of the specimens in the collection.
There are two proofs to think about in this packet.  The first is to do with the design of the calipers and the second is the appearance of the golden ratio in a 5 pointed star.  Students may not have time to tackle the proofs, but they can provide opportunities to extend this experience after the field trip, or can be considered if there is extra time.
Link to the Guide for Measuring Proportions (includes caliper proof):
Link to the Pentagon Proof Challenge:

Comparing Sequences (Mrs. Siegel).  In the breezeway, we’ll have a very non-traditional outdoor environment for working on a mathematical calculation.  This is intentional and we’ll see how it goes.  We’ll compare two sequences and derive the Golden Ratio.  We’ll also do “sequences” a little differently by setting up our stepping stone mats and playing the Fibonacci Stepping Stone Game.
Guide for Comparing Sequences- we’ll be using the quadratic formula!
Stepping Stone Game

Fibonacci Trees, Blocks and Bunnies (Mrs Lay)
Mrs. Lay, a teacher of both art and math will be in the small room in the visitors center opposite the cave.
She will guide the students in drawing some trees based on human and bee ancestry.  With these models constructed students can compare the two and see the Fibonacci pattern in the Bee Ancestry.  We will have the Fibonacci block puzzle and the Fibonacci Bunny Rabbit Problem models in this room as well.
Family Trees:
Fibonacci Building Blocks:
Bunny Problem

Fractal Sheet
The Fibonacci Trees, Hurricane Sandy, the Nautilus shell, and the Human and Bee Ancestry, all share characteristics with fractal forms.  The fractal sheet guides a pair of students – one to read the other to do through a couple of patterns and asks them to consider what is a fractal.  There will be some Fractal puzzles available in the Planetarium Room.
Fractal Sheet

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